network and model
Network meta-analysis provides a means of aggregating and comparing the results of all studies examining the effect of antidepressants on substantial pain.
We now define the parameters of the model for the substantial pain outcome, using White et al.’s notation.
Let \(i = 1, \dots, n\) denote the study, and \(k = 1, \dots, K\) denote the treatment.
Let \(R_i\) be the set of treatments in study \(i\), which we call the study design.
Let \(\theta_{ik}^a\) be the parameter of interest in arm \(k\) of study \(i\) (White et al. 2019).
Let \(y_{ik}^a\) denote the observed effect for arm \(k\) of study \(i\).
arm-based likelihood
Arm-based likelihood is
contrast-based likelihood
model results
pairwise analyses
Individual analyses for each treatment, compared with placebo.
- adapt notation to our problem
why do pairwise
pairwise meta-analysis
We want to fit a pairwise meta-analysis for each treatment, with concise summaries, as well as in-depth supplementary material. We split the analysis into two components, as the number of interventions per outcome is considerable, see below Table, in some cases, according to recommendations in Section 11.6.4 of the Handbook (Higgins et al. 2019).
| Number of studies and interventions, by outcome | ||
|---|---|---|
| Interventions include placebo, as sometimes there are more than one type of placebo | ||
| Outcome | Number of studies | Number of interventions |
| substantial pain | 4 | 5 |
| pain | 44 | 28 |
| mood | 61 | 42 |
pairwise model
To avoid confusion, we need to state the pairwise models being fun using the same notation as we use for the network meta-analysis models.
Here we consider a subset of data for \(j\), that is for only one outcome, \(j'\), so that we expect the same as above except we fix the index, so that \(y^a_{j = j'|ki}\).
\[ d_i \sim N(\delta_i, \sigma^2)\\ \delta_{i} \sim N(\delta, \tau^2) \] Given the data, what description of the data is most plausible, if we define the components of the data as follows? We assume a random-effects structure, which is to say, we can approximate, with a measurable degree of plausibility, the expected value of a standardised mean difference \(d_i\) of an outcome \(j'\) of interest for a study \(i\) and antidepressant treatment \(k\), as compared with placebo. We assume there is variability introduced by the study design \(\tau^2\), as well as sampling error \(\sigma^2\).
input: design matrix
A fixed effects NMA with a binomial likelihood (logit link).
Inference for Stan model: binomial_1par.
4 chains, each with iter=2000; warmup=1000; thin=1;
post-warmup draws per chain=1000, total post-warmup draws=4000.
mean se_mean sd 2.5% 25% 50% 75%
d[clomipramine] 0.46 0.00 0.26 -0.05 0.29 0.47 0.63
d[duloxetine] 0.72 0.00 0.19 0.35 0.59 0.71 0.84
d[mianserin] 0.51 0.00 0.26 0.00 0.33 0.50 0.68
d[trazodone] 0.42 0.01 0.52 -0.59 0.07 0.41 0.78
lp__ -644.48 0.05 1.99 -649.33 -645.60 -644.14 -643.03
97.5% n_eff Rhat
d[clomipramine] 0.96 2900 1
d[duloxetine] 1.08 2612 1
d[mianserin] 1.01 2899 1
d[trazodone] 1.46 2713 1
lp__ -641.57 1584 1
Samples were drawn using NUTS(diag_e) at Tue Apr 20 02:31:26 2021.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).output
A fixed effects NMA with a binomial likelihood (logit link).
Inference for Stan model: binomial_1par.
4 chains, each with iter=2000; warmup=1000; thin=1;
post-warmup draws per chain=1000, total post-warmup draws=4000.
mean se_mean sd 2.5% 25% 50% 75%
d[clomipramine] 0.46 0.00 0.26 -0.05 0.29 0.47 0.63
d[duloxetine] 0.72 0.00 0.19 0.35 0.59 0.71 0.84
d[mianserin] 0.51 0.00 0.26 0.00 0.33 0.50 0.68
d[trazodone] 0.42 0.01 0.52 -0.59 0.07 0.41 0.78
lp__ -644.48 0.05 1.99 -649.33 -645.60 -644.14 -643.03
97.5% n_eff Rhat
d[clomipramine] 0.96 2900 1
d[duloxetine] 1.08 2612 1
d[mianserin] 1.01 2899 1
d[trazodone] 1.46 2713 1
lp__ -641.57 1584 1
Samples were drawn using NUTS(diag_e) at Tue Apr 20 02:31:26 2021.
For each parameter, n_eff is a crude measure of effective sample size,
and Rhat is the potential scale reduction factor on split chains (at
convergence, Rhat=1).structuring nma data for metafor
concise summary
long-form supplementary